The Pythagorean Identity

30-60-90 triangle labeled sides a, b, c
a squared + b squared = c squared

In Geometry, you learned a very important theorem known as the Pythagorean Theorem. Now, we are going to apply the Pythagorean Theorem to verify some Trigonometry identities. So let's think about this in terms of sine and cosine.

If we let: equation image indicator LaTeX: x^2+y^2=1

Now, divide both sides by equation image indicator LaTeX: r^2 : $LaTeX: \frac{x^2+y^2}{r^2}=\frac{r^2}{r^2}$

And re-arranged a bit: equation image indicator $LaTeX: \frac{x^2}{r^2}+\frac{y^2}{r^2}=1$

Apply exponent rules: equation image indicator $LaTeX: \left(\frac{x}{r}\right)^2+\left(\frac{y}{r}\right)^2=1$

Replace with sine and cosine: equation image indicator $LaTeX: \left(\sin\theta\right)^2+\left(\cos\theta\right)^2=1,\:$ which can also be written this way: $LaTeX: \sin^2\theta+\cos^2\theta=1.$

This result is called a Pythagorean Identity. It can be written in different forms by manipulating the original equation. For example,

$LaTeX: \sin^2\theta+\cos^2\theta=1$

$LaTeX: \cos^2\theta=1-\sin^2\theta$

$LaTeX: \sin^2\theta=1-\cos^2\theta$

So now, we can use the Pythagorean Identity to verify identity problems.

On a separate sheet of paper, verify the identities below:

equation image indicator $LaTeX: 1.\:\left(\cos^2\theta-1\right)\left(\tan^2\theta+1\right)=-\tan^2\theta \\ 2.\:\frac{\left(\tan^2\theta+1\right)}{\tan^2\theta}=1+\cot^2\theta \\ 3.\:\tan B+\cot B=\csc B\sec B$

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